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Prove that the intersection of two planes which are not parallel is a line

Prove that if and are two planes which are not parallel then they intersect in a line.

Proof. Let the Cartesian equations of and be given by

respectively. Then, the intersection is given by the common solutions of these two equations. Since and are not parallel, we know they do not have the same normal vector so that for all . Further, since the normals are nonzero, we know each equation has at least one nonzero coefficient. Without loss of generality, let . Then,

Substituting into the Cartesian equation for we have

is the set of solutions for the points on . But, we know at least one of or is nonzero, otherwise . Hence, we have the equation for a line. Therefore, is a line

Prove a formula for the distance between a plane determined by three points and a point

1. If a plane is determined by the points prove that the distance from a point to this plane is given by

2. Using part (a) compute the distance in the case

1. Proof. We know the distance from a plane containing a point to a point not on the plane is given by the formula

Since the plane through is the set of points

we have . Thus, the distance from to is

2. For the given points we have

Find a Cartesian equation for a plane parallel to a given plane and equidistant from a given point

Consider a plane given by the equation

Find the Cartesian equation for a plane parallel to this one and the same distance as this plane from the point .

Since the requested plane is parallel to the given plane we know that they must have the same normal vector, . Therefore, the Cartesian equation of the requested plane is of the form

From the previous exercise (Section 13.17, Exercise #19) we know the distance from to a plane is given by the formula

Therefore, the distance from the given plane to the point is

Since the distance from the point to the requested plane must be the same we must have

(Since the solution belongs to the other plane.)

Prove some equations about distances between points and planes

1. Prove that the distance from the point to the plane

is given by the formula

2. Find the point on the plane which is nearest to the point .

1. Proof. By Theorem 13.6 (page 476 of Apostol) we know that the distance from a point to a plane is given by

2. A normal to the plane is given by . So, for any point . Further, the distance from to a point not on is minimal when where

Thus,

Naming to be the point we have

Prove that the intersection of a line and plane which are not parallel contains exactly one point

Prove that the intersection of a line and a plane such that the line is not parallel to the plane contains one and only one point.

Proof. Denote the line by and the plane by . Let be the set of points

Since is not parallel to w know that its direction vector is not in the span of and . Further, by definition of a plane, we know the vectors and are linearly independent. Hence, are linearly independent. Then, any point in the intersection must be a solution to the system of equations

By the linear independence of we know this system has exactly one solution . Hence, contains exactly one point

Find the parametric equation for a line through a point and parallel to two planes

We say that a line is parallel to a plane if the direction vector of the line is parallel to the plane. Let be the line containing the point and parallel to the planes

Find a vector parametric equation for .

The normal vectors of the planes are and . So, the direction vector of will be perpendicular to both of these,

From the first equation we have . Plugging this into the second equation we obtain , which then gives us . Since is arbitrary, we take to obtain a direction vector . Therefore, the vector parametric equation for the line is

Find the Cartesian equation of the plane parallel to given vectors with given intercept

Consider the plane which is parallel to both of the vector and and intersects the -axis at the point . Find the Cartesian equation of this plane.

Since contains the point and is parallel to and we have

Thus, the Cartesian equations for are

Hence,

Compute the volume of a tetrahedron with given vertices

Consider the tetrahedron with vertices at the origin and at the points where the plane

intersects the coordinate axes. Compute the volume of this tetrahedron.

First, the intercepts of the plane are given by . Then from a previous exercise (Section 13.14, Exercise #13) we know that the volume of a tetrahedron with vertices is

Letting we have

Find a Cartesian equation for a plane through a point with normal vector making given angles

Let be the plane whose normal vector makes angles with the unit coordinate vectors and which contains the point . Find a Cartesian equation for the plane.

Since the normal vector to the plane makes angles with the unit coordinate vectors, we have

Hence, . So, the plane has a Cartesian equation of the form

Since it contains we have . Therefore, a Cartesian equation of the plane is

Determine properties of a point whose movement in space is determined by a vector parametric equation

Consider a point moving in space with position at time given by

1. Prove that the motion of the point is along a line.
2. Find a vector parallel to this line.
3. Find the time at which the point intersects the plane with Cartesian equation .
4. What is the Cartesian equation for the plane parallel to the plane in part (c) which contains the point ?
5. Let be the plane perpendicular to containing the point . Find a Cartesian equation for .

1. Proof. We use the formula for the motion of the particle to compute

This is the parametric equation for the line through parallel to the vector

2. From part (a) we have a vector parallel to given by .
3. First, the line on which the point moves is the set of points

So, to find the intersection with the plane we compute

4. First, we have

Since we know the plane is parallel to the one in part (c) it has a Cartesian equation of the form

We compute . Hence, the plane has Cartesian equation

5. Since the plane is perpendicular to the line we know that it has a normal vector in the same direction as , so (from part (b)). Thus, we have a Cartesian equation of the form

Since the point

is on the plane we have . Therefore, the plane is given by